Let $\mathfrak{ r}$ be a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form, let $\mathfrak{ p}$ be a finite dimensional complex super vector space with a non-degenerate super-symmetric bilinear form, and let $\nu: \mathfrak{ r}\rightarrow\mathfrak{ osp}(\mathfrak{ p})$ be a Lie superalgebra homomorphism. In this paper, we give a necessary and sufficient condition for $\mathfrak{ r}\oplus\mathfrak{ p}$ to be a quadratic Lie superalgebra. The criterion obtained is an analogue of a constancy condition given by Kostant in the Lie algebra setting. As an application, we prove an analogue of the Parthasarathy's formula for the square of the Dirac operator attached to a pair of quadratic Lie superalgebras.
Categories: math.RT
